Stanley's proof in [1]} imports a result from complex geometry, namely the Hard-Lefschetz theorem, so for a while people attempted to find a more combinatorial proof, one that used more elementary arguments. McMullen succeeded initially in [2]} and then corrected and simplified parts of it in [3]}. The heart of the \(g\) -theorem lies in associating, to each simplicial polytope \(P\) , a finite dimensional algebra with particular properties. By now there are many different ways of doing this. For instance, in [4]} Billera uses piecewise polynomial functions on a fan (an approach also explored by Brion [5]}), and in [6]} Fulton-Sturmfels use the Chow ring of the associated toric variety which they show is isomorphic to the Minkowski weights, which is the same ring that McMullen employs in [3]} although he constructs it from his earlier work in [8]}.
